#509596

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(9,10,14,17)(11,21,23,39)(12,25,22,16)(13,29,32,38)(15,33,31,20)(18,24,36,34)(19,27,37,26)(28,35,30,40);(5,6)(7,8)(9,11,14,23)(10,16,17,25)(12,26,22,27)(13,30,32,28)(15,34,31,24)(18,37,36,19)(20,40,33,35)(21,38,39,29);(2,4)(6,8)(10,17)(16,25)(18,36)(19,37)(20,33)(21,39)(29,38)(35,40);(2,4)(9,12,27,11,15,28,13,24,14,22,26,23,31,30,32,34)(10,18,38,40,20,39,19,16,17,36,29,35,33,21,37,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416792);